# Quality engineering(QE) 19

This is the explanation material. →　QE19

p.1: Today's topic is "What is orthogonal?"

p.2: We can calculate the correlation coefficient immediately using Excel, but I would like you to understand the principle of the correlation coefficient. The x-axis is the percentage of rented houses, and the y-axis is the percentage of unmarried persons. The graph shows the correlation between x and y for the data in any region. "Covariance" is the average value of the sum of products of x and y calculated by subtracting the average value of x and y. Incidentally, the average value of the sum of squares obtained by subtracting the average value of x from each x is called "variance".

p.3: We calculate using correlation coefficient = (covariance of x and y)/{(standard deviation of x) x (standard deviation of y)}.  If the covariance is positive, the right shoulder rises, and if the covariance is negative, the right shoulder declines. Since variance = (standard deviation) 2, it is an image that the slope is calculated by dividing the covariance by (standard deviation of x) and (standard deviation of y).

p.4: Mathematically, it is expressed as x·y (inner product of vector x and y) = lxl·lylcos θ. The dot product of the vectors x and y is the covariance of p.3, and lxl and lyl are the lengths of the vectors x and y, which correspond to the standard deviation. The correlation coefficient r is cos θ obtained by dividing the inner product of the vectors by the length of each vector. If the vectors x and y match in the same direction, the correlation coefficient r=1 when θ=0, and the correlation coefficient r=0 when the vectors x and y are vertical θ=90° and cos90°=0. The correlation coefficient r=0 is "orthogonal". Think of the columns of each control factor in the orthogonal array as vectors. The orthogonal table is made so that the correlation coefficient of two columns (vector) will be 0 (zero).

p.5: Let's substitute the SN ratio of each control factor level into the allocation in the orthogonal table on the upper right. Calculating covariance and standard deviations 1 and 2 of arbitrary two columns such as B and C columns and C and D columns, the covariances are all 0 (zero), so the correlation coefficient is also 0 (zero). It is clear that it will be. In other words, all columns except the column of interest are orthogonal (no correlation). Using this property, we can independently evaluate which level is most effective for each column.